Changing Divine Smite over to dealing Necrotic Damage weakens the ability. In total, across all official sourcebooks, there are 9 creatures that are Resistant to Radiant Damage, 1 that is Immune, and 9 that are Vulnerable. On the other hand there are 46 creatures Resistant to Necrotic Damage, 38 that are Immune, and none that are Vulnerable. And, perhaps of note, many of the things Resistant or Immune to Radiant damage are often Good aligned--so adventurers are somewhat less likely to end up fighting them.
Thus, by switching the Paladin over to dealing Necrotic damage, you are increasing the number of creatures that can resist or ignore their Smite...and you've completely lost all creatures that are Vulnerable to your Smite.
On the other hand, if you allow them to switch between Radiant and Necrotic damage at-will, it is a (relatively mild) technical buff to the ability.
There are a total of 10 creatures that are Resistant or Immune to Radiant Damage...of those 10, 4 of them also have Resistance or Immunity to Necrotic Damage: Aurelia and a Deathpact Angel from the Guildmaster's Guide to Ravnica, a Solar from the Monster Manual, and Zariel from Mordenkainen's Tome of Foes. So, mechanically speaking, by allowing the Paladin to pick between Radiant and Necrotic, you are reducing the number of "totally immune" creatures from 1 to 0, and "Resistant" from 9 to 4.
As for Vulnerability...Divine Smite already deals Radiant Damage, which 9 creatures are Vulnerable to. Adding the option to switch to Necrotic damage does not add any more Vulnerable creatures--so that's effectively a non-issue. But this, of course, leads to...
While this is technically a buff, it will be limited by the character's knowledge. If you have the option to deal Necrotic damage, there's always the chance that you'll choose to use it against one of the 84 types of creature that are Resistant or Immune to Necrotic Damage. So while you technically have the mechanical advantage of being able to pick the right damage type to deal to get around Resistances and Immunities, or pick 'Radiant' against a creature that is Vulnerable to that type--there's a chance of picking the 'wrong one' and ending up hitting a Helmed Horror with Necrotic Damage (which it is immune to). In short--you're more likely to pick the 'wrong damage type' and end up dealing less damage than if you just stuck with always doing Radiant Damage.
(Note: totals are not including creatures that are simply named versions of a base monster...such as how dndbeyond lists several named creatures that are simply a name bolted to a Lich statblock. These totals were acquired by filtering on dndbeyond and then manually counting the results.)
Is this a Bad Thing?
Not necessarily. As long as the player knows what they are getting themselves into, if they are willing to take on a mechanical detriment to their character in the name of things fitting the design they had in their head--then that's fine.
I have played with and run games for many D&D players (and been one myself) who make mechanically sub-optimal choices in the name of building a character that fits the design they have in their head.